Where,
r is the radius
h is the height
l is the slant height
The area of the curved (lateral) surface of a cone = πrl
Note:
A cone does not have uniform (or congruent) cross-sections. (more about conic section here)
Example 1: A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.
Solution:
To begin with we need to find slant height of the cone, which is determined by using Pythagoras, since the cross section is a right triangle.
l2 = h2 + r2
l2 = 52 + 32
l2 = 25 + 9
l = √(34)
l = 5.83 cm
And the total surface area of the cone is:
SA = πr2 + πrl
SA = π · r · (r + l)
SA = π · 3 · (3 + 5.83)
SA = 83.17 cm2
Therefore, the total surface area of the cone is 83.17cm2
Example 2: The total surface area of a cone is 375 square inches. If its slant height is four times the radius, then what is the base diameter of the cone? Use π = 3.
Solution:
The total surface area of a cone = πrl + πr2 = 375 inch2
Slant height: l = 4 × radius = 4r
Substitute l = 4r and π = 3
3 × r × 4 r + 3 × r2 = 375
12r2 + 3r2 = 375
15r2 = 375
r2 = 25
r = 25
r = 5
So the base radius of the cone is 5 inch.
And the base diameter of the cone = 2 × radius = 2 × 5 = 10 inch.
Example 3: What is the total surface area of a cone if its radius = 4cm and height = 3 cm.
Solution:
As mentioned earlier the formula for the surface area of a cone is given by:
SA = πr2 + πrl
SA = πr(r + l)
As in the previous example the slant can be determined using Pythagoras:
l2 = h2 + r2
l2 = 32 + 42
l2 = 9 + 16
l = 5
Insert l = 5 we will get:
SA = πr(r + l)
SA = 3.14 · 4 · (4+5)
SA = 113.04 cm2
Example 4: The slant height of a cone is 20cm. the diameter of the base is 15cm. Find the curved surface area of cone.
Solution:
Given that,
Slant height: l = 20cm
Diameter: d = 15cm
Radius: r = d/2 = 15/2 = 7.5cm
Curved surface area = πrl
CSA = πrl
CSA =π · 7.5 · 20
CSA =471.24 cm2
Example 5: Height and radius of the cone is 5 yard and 7 yard. Find the lateral surface area of the given cone.
Solution:
Lateral surface area of the cone = πrl
Step 1:
Slant height of the cone:
l2 = h2 + r2
l2 = 72 + 52
l2 = 49 + 25
l = 8.6
Step 2: Lateral surface area:
LSA = πrl
LSA = 3.14 × 7 × 8.6
LSA =189.03 yd2
So, the lateral surface area of the cone = 189.03 squared yard.
Example 6: A circular cone is 15 inches high and the radius of the base is 20 inches What is the lateral surface area of the cone?
Solution:
The lateral surface area of cone is given by:
LSA = π × r × l
LSA =3.14 × 20 × 15
LSA = 942 inch2
Example 7: Find the total surface area of a cone, whose base radius is 3 cm and the perpendicular height is 4 cm.
Solution:
Given that:
r = 3 cm
h = 4 cm
To find the total surface area of the cone, we need slant height of the cone, instead the perpendicular height.
The slant height l can be found by using Pythagoras theorem.
l2 = h2 + r2
l2 = 32 + 42
l2 = 9 + 16
l = 5
The total surface area of the cone is therefore:
SA = πr(r + l)
SA = 3.14 · 3 · (3+5)
SA = 75.36 cm2